Finite Group Databasebeta

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#72317706

order1024
nilpotency class4
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C26

Aliases

No aliases for this group.

Subgroups

subgroupG/H
centerC4#8546
commutatorC22×#41C22×C4
Frattini, prime radical, cosocleC23×#41C23
socleC2#122051
LCSG ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ C23×#41 ◁ G
LECSG ▷ C23×#41 ▷ C23 ▷ C2 ▷ C1

2085 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 2085 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,11,7,12,26,39)(3,8,15,29,16,30,41,57)(4,17,20,50,21,35,25,56)(5,10,23,36,24,37,53,49)(9,31,33,19,34,54,38,22)(13,40,18,48,43,60,52,63)(14,44,46,64,47,28,59,61)(27,51,32,45,58,55,62,42);(1,3,6,15,7,16,26,41)(2,8,11,29,12,30,39,57)(4,18,25,43,21,52,20,13)(5,14,23,46,24,47,53,59)(9,32,38,58,34,62,33,27)(10,28,36,61,37,44,49,64)(17,48,56,60,35,63,50,40)(19,51,31,45,22,55,54,42);(1,4,6,25,7,21,26,20)(2,9,11,38,12,34,39,33)(3,13,41,52,16,43,15,18)(5,22,23,54,24,19,53,31)(8,27,57,62,30,58,29,32)(10,35,36,50,37,17,49,56)(14,45,59,51,47,42,46,55)(28,60,64,48,44,40,61,63)

Polycyclic presentation:

2^10|a^2=e,b^2=ef,c^2=efh,d^2=j,e^2=j,b^a=bg,c^a=cd,c^b=cefhj,d^a=dh,d^c=dj,e^b=ei,e^c=ehj,f^b=fi,f^c=fhj,g^a=gi,g^c=gj,g^d=gj,h^b=hj,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72317714, #72317737
maximal normal subgroup of#72317714, #72317737

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.5*F.6);;
gap> SetPower(coll, 3, F.5*F.6*F.8);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);